Asymptotic Properties for a General Class of Szász–Mirakjan–Durrmeyer Operators

Source abstract

ABSTRACT In this paper, we introduce a general family of Szász–Mirakjan–Durrmeyer type operators depending on an integer parameter . They can be viewed as a generalization of the Szász–Mirakjan–Durrmeyer operators, Phillips operators, and corresponding Kantorovich modifications of higher order. For , these operators possess the exceptional property to preserve constants and the monomial . It turns out that an extension of this family covers certain well‐known operators studied before, so that the outcoming results could be unified. We present the complete asymptotic expansion for the sequence of these operators. All its coefficients are given in a concise form. In order to prove the expansions for the class of locally integrable functions of exponential growth on the positive half‐axis, we derive a localization result, which is interesting in itself.